## Abstract

In this paper we investigate model-independent bounds for exotic options written on a risky asset using infinite-dimensional linear programming methods. Based on arguments from the theory of Monge–Kantorovich mass transport, we establish a dual version of the problem that has a natural financial interpretation in terms of semi-static hedging. In particular we prove that there is no duality gap.

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## Notes

For the sake of simplicity, we assume zero interest rate and no cash/yield dividends. This assumption can be relaxed by considering the process (

*f*_{ t }) introduced in [17] (see Eq. (14)) which has the property to be a local martingale.The cumulative distribution function of

*μ*_{ i }can be read off from the call prices through \(F_{i}(K)= 1 - \lim_{\varepsilon\downarrow0} 1/\varepsilon[ \mathcal{C}(t_{i},K)-\mathcal{C}(t_{i},K+\varepsilon) ]\) for*i*=1,…,*n*. Concerning the mathematical finance application it would be sufficient to consider strikes*K*≥0 and marginals which are concentrated on the positive half-line. We prefer to go with the more general case since the proofs are not more complicated. A technical difference is that call prices satisfy only \(\lim_{K\to-\infty}\mathcal{C}(t_{i},K) - K = s_{0}\) rather than the simpler \(\mathcal{C}(t_{i},0) = s_{0}\) in the case where*S*is assumed to be nonnegative.It might be expected that the delta strategy in (1.2) should also include a constant

*Δ*_{0}multiplier of (*s*_{1}−*s*_{0}) corresponding to an initial forward position. However, this term is not necessary as it can be subsumed into the term*u*_{1}.In more financial terms, this means that \(\mathcal{C}(t,K)\) is increasing in

*t*for each fixed \(K\in\mathbb{R}\).Most of the basic results are equally true for Polish probability spaces (

*X*_{1},*μ*_{1}),…,(*X*_{ n },*μ*_{ n }), but we do not need this generality here.We should like to emphasize that the lower/upper bounds corresponding to different strikes

*K*are attained by different martingale measures. This is not the case if we do not include the martingality constraint, as in this case the upper/lower bounds are attained by the co-monotone resp. anti-monotone coupling for each strike*K*(see for instance [36, Sect. 2.2.2]).In probabilistic terms, the measure \(\mathbb{Q}_{s_{1}}\) is the conditional distribution of

*S*_{2}under \(\mathbb{Q}\) given that*S*_{1}=*s*_{1}.We emphasize that while this simple guess works in the present setting, the situation is more subtle for general distributions.

Formally Hobson and Neuberger are interested to

*maximize*the price of the payoff |*S*_{2}−*S*_{1}| while we want to*minimize*the price of−|*S*_{2}−*S*_{1}|. Mathematically, the two problems are of course the same. We haven chosen the latter formulation to be consistent with the notation in our main result in Theorem 1.1.Some progress in this direction is made in [4, Appendix A]. (Note added in revision.)

I.e., \(g^{**}:\mathbb{R}\to\mathbb{R}\) is the largest convex function smaller than or equal to

*g*.

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## Acknowledgements

We thank the Associate Editor and the extraordinarily careful referees for their comments and in particular for pointing out a mistake in an earlier version of this article. We also benefitted from remarks by Johannes Muhle-Karbe.

The first author thanks the FWF for partial support through grant P21209. The third author acknowledges support from ERC grant No. 247033.

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## Appendix

### Appendix

As a special case of [28, Theorem 2.14], we have the duality equation

for every lower semi-continuous cost function \(\varPhi:\mathbb{R}^{n}\to [0,\infty]\). The main task in the subsequent proof of Proposition 2.1 is to show that the duality equation is obtained if one restricts to functions in the class \(\mathcal{S}\) in the dual problem.

### Proof of Proposition 2.1

As in the proof of Theorem 1.1, it is sufficient to prove the duality equation in the case *Φ*≥0.

Given a bounded continuous function *f* and *ε*>0, there is for every *i*=1,…,*n* some \(u\in\mathcal{S}\) such that *f*≥*u* and ∫(*f*−*u*)*dμ*
_{
i
}<*ε*. Therefore we may change the class of admissible functions from \(\mathcal{S}\) to \(C_{b}(\mathbb{R})\), i.e., it suffices to prove

We first show this under the additional assumption that \(\varPhi \in C_{c}(\mathbb{R}^{n})\). By [28, Theorem 2.14], there exist for each *η*>0 *μ*
_{
i
}-integrable functions *u*
_{
i
}, *i*=1,…,*n*, such that

and *u*
_{1}⊕⋯⊕*u*
_{
n
}≤*Φ*. Note that the latter inequality implies that *u*
_{1},…,*u*
_{
n
} are uniformly bounded since *Φ* is uniformly bounded from above.

To replace *u*
_{1} by a function in *C*
_{
b
}, we consider *H*=*Φ*−(*u*
_{1}⊕⋯⊕*u*
_{
n
}) and define

for \(x_{1}\in\mathbb{R}\). We claim that \(\tilde{u}_{1}\) is (uniformly) continuous. Indeed, as *Φ* is uniformly continuous, for every *ε*>0 there exists *δ*>0 such that whenever \(x,x'\in \mathbb{R}\), |*x*−*x*′|<*δ*, then

Thus we obtain

whenever |*x*−*x*′|<*δ*. By definition \(\tilde{u}_{1}\) is also bounded from below and satisfies \(\tilde{u}_{1}\geq u_{1}\) as well as

Iteratively replacing the functions *u*
_{2},…,*u*
_{
n
} in the same fashion, we obtain (A.1) in the case \(\varPhi\in C_{c}(\mathbb{R}^{n})\).

Finally, the same argument as in the proof of Theorem 1.1 gives the duality relation in the case of a general, lower semi-continuous function \(\varPhi:\mathbb{R}^{n}\to[0,\infty ]\). □

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Beiglböck, M., Henry-Labordère, P. & Penkner, F. Model-independent bounds for option prices—a mass transport approach.
*Finance Stoch* **17**, 477–501 (2013). https://doi.org/10.1007/s00780-013-0205-8

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DOI: https://doi.org/10.1007/s00780-013-0205-8

### Keywords

- Model-independent pricing
- Monge–Kantorovich transport problem
- Option arbitrage
- Robust superreplication theorem